Fuchsian Holonomic Sequences∗

Fuchsian holonomic sequences∗ JORIS VAN DER HOEVEN CNRS, École polytechnique, Institut Polytechnique de Paris Laboratoire d'informatique de l'École polytechnique (LIX, UMR 7161) 1, rue Honoré d'Estienne d'Orves Bâtiment Alan Turing, CS35003 91120 Palaiseau, France Email: [email protected] Many sequences that arise in combinatorics and the analysis of algorithms turn out to be holonomic (note that some authors prefer the terminology D-finite). In this paper, we study various basic algorithmic problems for such sequences ( fn)n∈ℕ: how to compute their asymptotics for large n? How to evaluate fn efficiently for large n and/or large precisions p? How to decide whether fn >0 for all n? n We restrict our study to the case when the generating function f = ∑n∈ℕ fn z satisfies a Fuchsian differential equation (often it suffices that the dominant singularities of f be Fuchsian). Even in this special case, some of the above questions are related to long-standing problems in number theory. We will present algorithms that work in many cases and we carefully analyze what kind of oracles or conjectures are needed to tackle the more difficult cases. 1. INTRODUCTION 1.1. Statement of the problems ℕ Let 핂 be a subfield of ℂ. A sequence ( fn)n∈ℕ ∈ ℂ is said to be holonomic over 핂 if it satisfies a difference equation Σs(n) fn+s + ⋅ ⋅ ⋅ +Σ0(n) fn = 0, (1.1) s where Σ=Σs휎 +⋅⋅⋅+Σ0∈핂[n][휎] is a linear difference operator in 휎:n↦n+1 with Σs≠0. (Note that some authors prefer the terminology D-finite or P-finite instead of holonomic.) Many interesting sequences from combinatorics, the analysis of algorithms, and number theory are holonomic [42, 11]. We say that ( fn) is 핂-holonomic if fn ∈ 핂 for all n ∈ ℕ. We say that the equation (1.1) is non-degenerate if Σs(n) ≠ 0 for all n ∈ ℕ. In that case, the sequence is entirely determined by its first s coefficients and ( fn) is 핂-holonomic s if and only if ( f0, . ., fs−1)∈핂 . Throughout this paper, we assume that 핂 = ℚalg is the field of algebraic numbers. ℕ Given a non-degenerate 핂-holonomic sequence ( fn)∈핂 , one may raise several natural questions: Q1. How to compute the asymptotic expansion of fn when n→∞? Q2. What kind of constants coefficients can occur in the asymptotic expansion of fn? Q3. How to compute terms fn of the sequence efficiently as a function of n? Q4. How to decide whether fn > 0 or fn ⩾ 0 for large, all, or infinitely many n ∈ ℕ? alg (For this question, we assume that fn ∈ℚ ∩ ℝ for all n.) ∗. This article has been written using GNU TEXMACS [23]. 1 2 FUCHSIAN HOLONOMIC SEQUENCES These questions are related and can be further refined. For instance, it is natural to compute terms fn as elements of 핂. However, if n becomes large, then the bit-size of fn is generally at least proportional to n. When that happens, it may be preferable to switch to a floating point representation. If we have an asymptotic expansion of fn with suitable error bounds, then we may exploit that to quickly compute floating point approximations of the fn for large n. Similarly, if the dominant term of the expansion of fn is positive and provably dominates the other terms, then we may deduce that fn >0 for large n. Assuming that the generating function n f (z) ≔ fn z n∈ℕ is analytic at the origin, it is well known that the asymptotic behavior of the sequence ( fn) is closely related to the behavior of f near its dominant singularities (i.e. the singularities of smallest absolute value). As will be recalled in section 2.1, the generating function f is again holonomic, in the sense that it satisfies a non-trivial linear differential equation with polynomial coefficients. Holonomic functions can be evaluated extremely efficiently and their singularities are well understood. In this paper we will restrict our attention to the special case when at least the dominant singularities of f are Fuchsian (see section 2.2 for a definition). In that case, the behavior of f near its dominant singularities becomes much simpler and the evaluation of f near these singularities even more efficient. In their full generality, the questions Q1, Q3, and Q4 turn out to be very difficult, even in the Fuchsian case. Indeed, if f is actually a rational function, then the last question Q4 is related to Skolem's problem [30]. More precisely, Skolem's problem asks whether fn =0 2 n for some n ∈ ℕ. Now if f is a rational function, then so is ∑n∈ℕ fn z , so this question reduces to testing whether fn > 0 for all n ∈ ℕ. The hard cases for Skolem's problem occur when f has several dominant singularities. One archetype example is fn = 휆cos(훼n) +휇 cos(훽n) +휈 cos(훾n), (1.2) with ei훼, ei훽, ei훾, 휆, 휇, 휈 ∈ 핂 ∩ ℝ. A variant of this problem is also relevant for the question Q3: if certain terms of this sequence (1.2) can become “absurdly small”, then the computation of floating point approximations for these terms may take much longer than expected, since we need to compute with a precision that exceeds the order of cancellation. On the positive side, examples like (1.2) are fairly pathological, so it remains reason- able to hope answering our questions for most practical examples from combinatorics or the analysis of algorithms. One interesting concrete example was studied in [33]. The authors exploit the fact that the then open problem about the uniqueness of the Canham model for biomembranes reduces to proving the positivity of a certain holonomic sequence. They solve the latter problem using singularity analysis [11] and techniques for reliable numerical computations with holonomic functions [8, 18, 19, 34, 35]. In the present paper, among other things, we will extend this approach to more general holonomic sequences. Note that other approaches to automatically prove the positivity of sequences were proposed in [15, 29]. In fact, the main purpose of this paper is to provide the best possible answers to questions Q1–Q4 as long as we do not run into number theoretic trouble. We will also identify the precise nature of possible trouble, thereby clarifying which problems need to be over- come if we want to give even better answers. Most of our results rely on well known JORIS VAN DER HOEVEN 3 techniques. Our main contribution is therefore a detailed analysis of how to answer the questions Q1–Q4 as well as possible using these techniques. 1.2. Overview of our contributions Let us briefly outline the structure of this paper. Section 2 contains reminders about holonomic functions, Fuchsian singularities, and holonomic constants. In section 3, we start with questions Q1 and Q2. In order to obtain asymptotic expan- sions, we use Cauchy's classical contour integral formula for fn and deform the contour into a finite number of loop integrals around the smallest singularities of f (section 3.1). Each of these loop integrals is a truncated Mellin-style integral, whose asymptotic expansion can be computed using classical formulas (section 3.2). (In the context of difference equations, note that some authors use the terminology “Pincherle transform” [39] instead of “Mellin transform”.) Adding up the contributions from each of the singularities, we obtain an asymptotic expansion for fn (Theorem 3.2). The coefficients of this asymptotic expansion can be computed explicitly and expressed in terms of (non-singular) holonomic constants and values of higher derivatives of 훾(z) = Γ(z)−1 at points in 핂. Using reliable numeric techniques from [19, 33], one may also compute explicit bounds for the error of the asymptotic expansion (section 3.3). Unfortunately, Theorem 3.2 is imperfect in the sense that some of the terms in the “asymptotic expansion” for fn may cancel out (in which case the bound for the error becomes larger than the expansion itself). This may actually happen in three different ways that will be analyzed in detail in section 4. ¯ First of all, consider a holonomic function g that converges on the closed unit disk 풟1. Its value g(1) at z=1 is a holonomic constant (and any non-singular holonomic constant can be obtained in this way). Now f =g/(1−z) is also a holonomic function and the first term of the asymptotic expansion of fn is given by fn ∼g(1) if and only if g(1) ≠ 0. This shows that we need a zero-test for holonomic constants in order to detect cancellations of terms in the asymptotic expansion of fn. A second kind of cancellation may occur between distinct terms in the asymptotic expansion and only for certain values of n. Consider for example 1 1 1 f (z) = + + 1− z 1 +z 1 −z/2 n n −n fn = 1 + (−1) + 2 . Then the dominant terms 1n and (−1)n cancel out for odd values of n. We call this phenomenon “resonance” and the good news is that it can be entirely eliminated by considering a finite number of cases in which fn is replaced by a subsequence fΠn+휌 with Π ∈ℕ> and 휌 ∈{0, . .,Π −1} (see section 4.2). A non-periodic variant of resonance is “quasi-resonance”. Consider for instance the sequence ( fn)n∈ℕ from (1.2). We say that this sequence is quasi-resonant if, for every −휅 C> 0 and 휅 ∈ ℝ, there exist an infinite number of n >0 with | fn| >Cn .

Load more